Let's analyze the given work for both Holly and Tamar to determine the accuracy of their solutions.
### Holly's Work
Holly attempts to verify if:
[tex]\[ (40 + 42)^2 = 58^2 \][/tex]
Calculating both sides:
- On the left side of the equation:
[tex]\[ (40 + 42)^2 = 82^2 \][/tex]
[tex]\[ 82^2 = 82 \times 82 = 6,724 \][/tex]
- On the right side of the equation:
[tex]\[ 58^2 = 58 \times 58 = 3,364 \][/tex]
Since:
[tex]\[ 6,724 \neq 3,364 \][/tex]
Holly's conclusion should be:
[tex]\[ \text{The triangle is not a right triangle.} \][/tex]
### Tamar's Work
Tamar attempts to verify if:
[tex]\[ 42^2 + 40^2 = 58^2 \][/tex]
Calculating both sides:
- On the left side of the equation:
[tex]\[ 42^2 + 40^2 = 42 \times 42 + 40 \times 40 \][/tex]
[tex]\[ 42^2 = 1,764 \][/tex]
[tex]\[ 40^2 = 1,600 \][/tex]
[tex]\[ 42^2 + 40^2 = 1,764 + 1,600 = 3,364 \][/tex]
- On the right side of the equation:
[tex]\[ 58^2 = 58 \times 58 = 3,364 \][/tex]
Since:
[tex]\[ 3,364 = 3,364 \][/tex]
Tamar's conclusion should be:
[tex]\[ \text{The triangle is a right triangle.} \][/tex]
### Conclusion
Based on the calculations:
- Holly's method is incorrect because the equation she used does not hold true: [tex]\( (40 + 42)^2 \neq 58^2 \)[/tex].
- Tamar's method is correct because the equation she used holds true: [tex]\( 42^2 + 40^2 = 58^2 \)[/tex].
Hence, Tamar is correct.
